Theorem imai 5058

Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
imai	⊢ ( I “ 𝐴) = 𝐴

Proof of Theorem imai

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfima3 5045	. 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2		df-br 4061	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3		vex 2780	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	ideq 4849	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5	2, 4	bitr3i 186	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
6	5	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦))
7		ancom 266	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
8	6, 7	bitri 184	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
9	8	exbii 1629	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
10		eleq1 2270	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
11	3, 10	ceqsexv 2817	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
12	9, 11	bitri 184	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦 ∈ 𝐴)
13	12	abbii 2323	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴}
14		abid2 2328	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
15	1, 13, 14	3eqtri 2232	1 ⊢ ( I “ 𝐴) = 𝐴

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2178  {cab 2193  ⟨cop 3647   class class class wbr 4060   I cid 4354   “ cima 4697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-br 4061  df-opab 4123  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707

This theorem is referenced by:  rnresi  5059  cnvresid  5368  ecidsn  6694